Formalization of Quantum Mechanics

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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In the formalism of quantum mechanics, the probability that an electron in the initial state is transferred into the final state, is given by Fermi s Golden Rule ... 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

An isolated microscopic system is fully determined, in the quantum mechanical sense, when its state function y/ is known. In the Dirac formalism of quantum mechanics, the state function can be identified with a vector of state, 11//). (11) The system in the 1y/ state may equivalently be described by a Hermitian operator, the so-called density matrix p of a pure state,... 

In the formalism of quantum mechanics, observables are associated to hermitian operators that act on the Hilbert space of square integrable functions representing the state of the quantum system. In the following, for the sake of definiteness, we shall consider hermitian operators B which can be written as hermitain combinations of position and momentum operators,... 

Hence, the chemical structure, dynamics, and spectroscopy of single molecules cannot be rigorously discussed in the present formalism of quantum mechanics and the problem is to construct a quantum theory for individual molecules. One possible starting point is an averaged (e.g., thermal) description and an averaged dynamics over an ensemble of molecules. The average ranges over all the pure states of the molecules in the ensemble. In technical terms, an individual quantum theory is related to a decomposition of nonpure states into pure ones (for a precise definition, see subsequent text). This decomposition is not unique. One... 

Some interesting behavior in single-molecule spectroscopy involves the stochastic migration of lines. Usual statistical quantum theory describes only mean values or dispersions of observables, but not the actual fluctuations in the dynamics of single quantum systems. In an individual formalism of quantum mechanics, such fluctuations are of great importance. 

IV. STATISTICAL VERSUS INDIVIDUAL FORMALISMS OF QUANTUM MECHANICS... 

In summary, statistical quantum mechanics permits us to derive strictly classical observables (such as the classical specific magnetization operator) by appropriate limit considerations (such as a limit of infinitely many spins in case of the Curie-Weiss model). However, statistical quantum mechanics cannot cope with fuzzy classical observables (for finitely many degrees of freedom) since different decompositions of a thermal state Dp are considered to be equivalent. The introduction of a canonical decomposition of Dp into pure states will give rise to an individual formalism of quantum mechanics in which fuzzy classical observables can be treated in a natural way. 

Let us now again consider single ammonia-type molecules and first review the problem of the usual statistical formalism of quantum mechanics (which uses density operators and no particular decompositions of... 

A typical question which makes no obvious sense in the usual statistical formalism of quantum mechanics is whether there exists some change in behavior in the sequence (monodeuteroaniline ammonia ... 

Let us now turn to the individual formalism of quantum mechanics again, where thermal states are decomposed in a canonical way according to the maximum entropy principle. We cannot compute yet maximum entropy decompositions and large-deviation entropies for molecular situations. Nevertheless, the (simpler) example of the Curie-Weiss magnet suggests that even in molecular situations ... 

In this chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system of postulates which will be the basis of our subsequent applications of quantum mechanics. 

These rules are a part of the formalism of quantum mechanics that is recovered from Schwinger s action principle. 

In the formalisms of quantum mechanics the basic problem has been discussed. Today these problems are understood to the extent that we have at our disposal a formulation of the logic of quantum mechanics (8.9). 

We remark that this is an exact correspondence, which is just a precise expression of Dirac s transformation formalism of quantum mechanics [14,15]. The explicit writing of the projector Pn in Eqs. (28a)-(28c) is motivated by the fact that in this way the operators //q(x) Ph, lVSje, aSje, 4e> an(J 316 aEo well-defined in the total space jSf (S1, t/0/2xc), and the discussion of the limit n > oo becomes conceptually clearer. 

Quantum mechanics is very much more than just a theory it is a completely new way of looking at the world, involving a change of paradigm perhaps more radical than any other in the history of human thought" [Leggett 2002], However, the linear formalism of quantum mechanics extrapolated from the level... 

The true philosophical import of the statistical interpretation has already been explained in 7 (p. 82). It consists in the recognition that the wave picture and the corpuscle picture are not mutually exclusive, but are two complementary ways of considering the same process—a process whose accessibility to intuitive apprehension is never complete, but always subject to certain limitations given by the principle of uncertainty. Here we have only one more important point to mention. The uncertainty relations, which we have obtained simply by contrasting with one another the descriptions of a process in the language of waves and in that of corpuscles, may also be rigorously deduced from the formalism of quantum mechanics—as exact inequalities, indeed for instance, between the co-ordinate q and momentum p we have the relation... 

XXII. The Formalism of Quantum Mechanics, and the Uncertainty Relation (pp. 84, 135). 

The deduction of the uncertainty relation, with the help of diffraction phenomena and other processes capable of being visualized, gives only a result specifying an order of magnitude. To obtain an exact inequality, we must call upon the formalism of quantum mechanics. Of this a short exposition will now be given, and the various theorems will be illustrated by examples in the preceding Appendices. 

To extract information from the wavefunction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfunctions. An extensive discussion of these important elements of the formalism of quantum mechanics is precluded by space limitations. For further details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide information about the system. The third postulate relates to certain properties of these operators ... 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

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